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Global radial basis function (RBF) collocation methods with inifinitelysmooth basis functions for partial differential equations (PDEs) work ingeneral geometries, and can have exponential convergence properties for smoothsolution functions. At the same time, the linear systems that arise are denseand severely ill-conditioned for large numbers of unknowns and small values ofthe shape parameter that determines how flat the basis functions are. We useHelmholtz equation as an application problem for the theoretical analysis andnumerical experiments. We analyse and characterise the convergence propertiesas a function of the number of unknowns and for different shape parameterranges. We provide theoretical results for the flat limit of the PDE solutionsand investigate when the non-symmetric collocation matrices become singular. Wealso provide practical strategies for choosing the method parameters andevaluate the results on Helmholtz problems in a curved waveguide geometry.

An investigation of global radial basis function collocation methods applied to Helmholtz problems